Pankration - The Art of ZvZ; Part 1

15hatch
can be followed by:
-pool/gas
-gas/pool
-pool; gasless
-pool with later gas

Pros:
-strong economy
-faster creep at natural
-faster queen production

Cons:
-prone to early pools:
-9pool will at least force you to cancel your natural expansion
-6 and 8pool can kill you (depends on your pool timing and worker control)
-10pool bane becomes a real threat if you can't micro at all
-later zerglings mean later scout on 3 and 4 player maps

With the advantages of 15hatchery significantly outweighing the downsides, it shouldn't be much of a surprise that this build is the favorite of progamers. In hands of a good player, 15hatch becomes safe vs every build, only really being at a disadvantage against well executed 9 and 10pools.
Faster queen production and earlier creep at natural offer a good set up for gasless play styles that relly on early queens and wall offs to defend.
While you sometimes want the safety of pool first (especially on 3 and 4 player maps where you can't always scout enemy rush with OL), 15hatchery has become the golden standard of high level ZvZ and for a good reason.

15pool
can be followed by
-hatchery/gas
-gas/hatchery
-hatchery; gasless
-hatchery with later gas

Pros:
-a build order win vs any kind of early pool
-faster scout
-able to deny OL scout with early queen

Cons:
-slow economy if you didn't opt for gasless
-later creep at natural
-late 2nd queen

Ah, the always so scared brother of 15hatch. 15pool is another bread&butter build in ZvZ, but a bit less popular amongst Korean. However, pool first is still perfectly to play at any level below the top. It sees a lot more popularity amongst foreigners (notably Nerchio, who very very rarely deviates from 15pool and is one of the best ZvZ players outside Korea) and while behind 15hatch in early game, you just need a single good exchange to even the things out and the better player should have no problem at all winning the game after opening 15pool even vs the greedier versions of hatchery first.
IMHO, the biggest setback of going 15pool vs 15hatch is losing the initiative after your first 2 lings, since the 15hatch player will have his larva production kick in earlier.

Early pools in ZvZ are aimed to keep a couple things in check:
-greedy hatchery first openings
-lazy overlord scouting patterns

Now, to go to the actual early pools, there's a few of these:

9pool:
good versus: 15hatch, 8pool, 7pool, 6pool
bad versus: 15pool, 14/14, 13/12, 10pool speed, 10pool bane

The goal of 9pool is to punish a 15hatchery opening, the 9pooler will try to either go for the drones or the hatchery of their opponent and give himself a lead through an ultimately earlier expansion (after the cancel) or more workers (after the attack). 9pool is not destined to end the game, but rather secure an advantage over hatchery first.
If 9pool meets any pool timing between 15pool and 10pool, it's kind of a build order loss, sorry.

8pool:
good versus: 15hatch, 7pool, 6pool
bad versus: 15pool, 14/14, 13/12, 10pool speed, 10pool bane, 9pool

It's kinda like 9pool, you hope for a hatch first. With 8pool, you will be looking for a locked kill with your lings, not a lead. 8pool will very often also feature a drone to spine rush opponent''s main base outside of his vision on creep.

10pool bane:
good versus: literally nothing, Tasteless (actually good vs all other early pools and 10pool speed)
bad versus: 15pool, 14/14, 13/13, properly executed 15hatch
A build that is nowadays considered outdated at the pro level, instead of going for an earlier pool or economic followup, you just make banes and try to nuke all of opponents drones.
Isn't considered viable since Leenock showed us how 15hatch can destroy 10pool bane with good drone micro. A notable player who keeps falling for 10pool bane is Life (Watch Dark vs Life from Kespa Cup). Still a rather good build on amateur level.
10pool bane also fares very well vs other early pools because of being able to morph defensive banelings

10pool and 10pool expand
good versus: bad players
bad versus: everyone else
nowadays more of a curiosity than an actual build, but it can be done in ZvZ and is worth mentioning.
the 10pool expand build goes like this: 10pool, double extractor trick, 12overlord, 6 lings, then drones
10pool is like 9pool, you aim to cancel the natural/cripple and macro behind it. On pro level, 10pools are very easily defendable with hatchery first so they have been retired. Still work on ladder, because people don't know the timings of early pools that well. Especially 10pool expand that can throw off even 15pool, as people won't notice the subtle differences in ling timings and will overestimate their economic lead

6pool and 7pool
good versus: bad overlord scouting paths
bad versus: literally every build that isn't hatchery first

6pool sucks, 7pool is even worse. We've seen I think 2(?) 6pools in high level ZvZ. This build is simply meant to abuse a playing going hatchery first who you know won't scout your early drone/lings with OL and just die when the lings arrive and you have a spine in their base. Like in this game. Jaedong 6pools because 1st ovie on Newkirk is sent not across the map, but across the dead space. He wins.

These builds aren't really early pools, but they still mostly rely on the element of surprise to take full effect. Very often used on 3 and 4player maps where your initial overlords aren't guaranteed to scout your opponent on the first try.

14/14
14gas 14pool, speedling all-in. Almost non-existant at high level. You just kind of make lings with this one, sometimes fake an expansion (Miniraser beat Snute with a build like this) and then go for the throat, hoping to abuse lackluster defence or subpar micro of your opponent.

13/12
13gas 12pool, speedling/baneling all-in. Created by Life himself, this build was designed for 4p maps. By the time your opponent can scout you (provided he didn't just scout you 1st with his overlord, then you lose the coinflip and lose, sorry), you have lings ready to deny scouting any scouting. If you succeed at it, you can just straight up kill anyone who even very slightly underprepares for the incoming attack.
Now as the pros have learned how to play vs this, the build has been retired, but is still very good for the ladder

10pool speed
Ok this one is both an early pool and 1 base speedling all-in.
You can execute it both ways:
-you send first lings to your opponent's base, cause some havoc, disrupt his build, then you catch him off-guard with a swell of speedlings and he dies
-you don't send your first lings to harass, just keep pooling units until speed is nearing completion, then leave your base to attack and (with a bit of scouting luck) he sees you at the last possible moments, has nothing and dies. Here's Jaedong doing this variant to sucker punch Hydra (who scouts it and still loses).